GROUP ACTIONS IN A REGULAR RING

On dihedrants admitting arc-regular group actions

We consider Cayley graphs Γ over dihedral groups, dihedrants for short, which admit an automorphism group G acting regularly on the arc set of Γ . We prove that, if D2n ≤G≤ Aut(Γ ) is a regular dihedral subgroup of G of order 2n such that any of its index 2 cyclic subgroups is core-free in G, then Γ belongs to the family of graphs of the form (Kn1 ⊗ · · · ⊗Kn )[Kc m], where 2n= n1 · · ·n m, and...

Ring Theory, Group Actions, Evolutionary Algorithms, and Global Optimization

In this paper we explain, using dynamical systems analysis, how evolutionary algorithms (EAs) solve global optimization problems. We start with a basic definition of EAs and a baseline random heuristic. Vose and Wright’s 1998 generalization of the Vose simple evolutionary algorithm model allows us to characterize alphabets of different lengths as multiplication and addition over rings of intege...

Rigid Resolution of a Finitely Generated Module Over a Regular Local Ring

The Maximal Regular Ideal of a Ring

Group Actions

The symmetric groups Sn, alternating groups An, and (for n ≥ 3) dihedral groups Dn behave, by their very definition, as permutations on certain sets. The groups Sn and An both permute the set {1, 2, . . . , n} and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If ...